Download [PDF] 2014 UKMT Senior Kangaroo Questions Mathematical Challenge United Kingdom Mathematics Trust

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UKMT UKMT UKMT SENIOR ‘KANGAROO’ MATHEMATICAL CHALLENGE Friday 28th November 2014 Organised by the United Kingdom Mathematics Trust The Senior Kangaroo paper allows students in the UK to test themselves on questions set for the best school-aged mathematicians from across Europe and beyond. RULES AND GUIDELINES (to be read before starting): 1. Do not open the paper until the Invigilator tells you to do so. 2. Time allowed: 1 hour. 3. The use of rough paper is allowed; calculators and measuring instruments are forbidden. 4. Use B or HB pencil only to complete your personal details and record your answers on the machine-readable Answer Sheet provided. All answers are written using three digits, from 000 to 999. For example, if you think the answer to a question is 42, write 042 at the top of the answer grid and then code your answer by putting solid black pencil lines through the 0, the 4 and the 2 beneath. Please note that the machine that reads your Answer Sheet will only see the solid black lines through the numbers beneath, not the written digits above. You must ensure that you code your answers or you will not receive any marks. There are further instructions and examples on the Answer Sheet. 5. The paper contains 20 questions. Five marks will be awarded for each correct answer. There is no penalty for giving an incorrect answer. 6. The questions on this paper challenge you to think, not to guess. Though you will not lose marks for getting answers wrong, you will undoubtedly get more marks, and more satisfaction, by doing a few questions carefully than by guessing lots of answers. Enquiries about the Senior Kangaroo should be sent to: Maths Challenges Office, School of Maths Satellite, University of Leeds, Leeds, LS2 9JT Tel. 0113 343 2339 www.ukmt.org.uk

1. Three standard dice are stacked in a tower so that the numbers on each pair of touching faces add to 5. The number on the top of the tower is even. What is the number on the base of the tower? 2. How many prime numbers have the property that is also prime?pp 4+1 3. Neil has a combination lock. He knows that the combination is a four-digit number with first digit 2 and fourth digit 8 and that the number is divisible by 9. How many different numbers with that property are there? 4. In the diagram, triangle is isosceles with and point lies on with and . What is the size in degrees of angle ?ABC C A=CB D AB AD=AC DB=DC BC A AB C D 5. Six of the seven numbers 11, 20, 15, 25, 16, 19 and 17 are divided into three groups of two numbers so that the sum of the two numbers in each group is the same. Which number is not used? 6. The numbers , and satisfy the equations and . What is the value of ? xy z x 2yz 3=7 3 xy 2=7 9 xyz 7 7. A table of numbers has 21 columns labelled 1, 2, 3, ..., 21 and 33 rows labelled 1, 2, 3, ..., 33. Every element of the table is equal to 2. All the rows whose label is not a multiple of 3 are erased. All the columns whose label is not an even number are erased. What is the sum of the numbers that remain in the table? 8. Andrew wishes to place a number in each circle in the diagram. The sum of the numbers in the circles of any closed loop of length three must be 30. The sum of the numbers in the circles of any closed loop of length four must be 40. He places the number 9 in the circle marked . What number should he put in the circle marked ?XY X Y

9. Each of the cubes in the diagram has side length 3 cm. The length of is cm. What is the value of ?AB k k AB 10. A Mathematical Challenge consists of five problems, each of which is worth a different whole number of marks. Carl solved all five problems correctly. He scored 10 marks for the two problems with the lowest numbers of marks and 18 marks for the two problems with the highest numbers of marks. How many marks did he score for all five problems? 11. The mean weight of five children is 45 kg. The mean weight of the lightest three children is 42 kg and the mean weight of the heaviest three children is 49 kg. What is the median weight of the children in kg? 12. On Old MacDonald's farm, the numbers of horses and cows are in the ratio 6:5, the numbers of pigs and sheep are in the ratio 4:3 and the numbers of cows and pigs are in the ratio 2:1. What is the smallest number of animals that can be on the farm? 13. The diagram shows a circle with diameter . The coordinates of are and the coordinates of are . The circle cuts the -axis at points and . What is the length of ?AB A(−2, 0) B(8, 0)yDE DE ABD E(8, 0) (−2, 0) 14. Rachel draws 36 kangaroos using three different colours. 25 of the kangaroos are drawn using some grey, 28 are drawn using some pink and 20 are drawn using some brown. Five of the kangaroos are drawn using all three colours. How many kangaroos did she draw that use only one colour?

15. A box contains seven cards numbered from 301 to 307. Graham picks three cards from the box and then Zoe picks two cards from the remainder. Graham looks at his cards and then says “I know that the sum of the numbers on your cards is even”. What is the sum of the numbers on Graham's cards? 16. The numbers , and satisfy the equations and . What is the value of ?xy z x+y+z=15 1 x+ 1 y+ 1 z=0 x 2+y 2+z 2 17. In the diagram, is a square. is the midpoint of . The area of the square is times the area of the shaded region. What is the value of ?PQRS M PQ k k P QR S M 18. Twenty-five workmen have completed a fifth of a project in eight days. Their foreman then decides that the project must be completed in the next 20 days. What is the smallest number of additional workmen required to complete the project on time? 19. In the long multiplication sum shown, each asterisk stands for one digit. What is the sum of the digits of the answer? 20. In the quadrilateral with cm and cm, point lies on so that cm and so that is parallel to . What is the length in centimetres of ? PQRS PQ=PS=25QR=RS=15TPQ PT=15TS QR TS m n op q cm 15 cm 15 cm 15 cm 25